I believe if you have beats you wont get the the 3 months free from what I have read if you want to be able to transfer your playlists over.
You can start a new account, you prob need to make sure your beats account email and apple account email don't match but I'm not 100% sure, and you will get 3 months free but then you wont be able to transfer playlists over. Again this is something that I have read.
My worry is I want to go from beats fam plan to a apple music fam plan. But due to the how the two apps work together by not showing the fam plan as an option after clicking get started with apple music that my beats account wont get canceled and playlists wont transfer. Because the way I read it in the app is it only transfers to individual plans maybe. So if you happen to be transferring a fam plan to fam plan please let me know how it goes.